Protractor, an instrument for laying off angles in plotting. There are four principal forms of the protractor: the rectangular, the semicircular, the circular, and the reflecting. The rectangular consists usually of a thin rectangular piece of ivory or metal, three edges of which are graduated from 0 to 180 degrees by portions of radii converging to the middle of the fourth edge as a centre;it is used only where a loose approximation to accuracy suffices. The circular and semicircular protractors, with either two, one, or no arms, are graduated circular arcs (usually metal), with or without flat straight-edged arms, turning about their perforated centres, and carrying verniers for the accurate reading of their arcs. But as they are only capable of protracting and measuring single angles on a map, they have not so wide a range of usefulness in engineering and surveying as the three-arm protractor. The three-arm circular protractor is a modification of the station-pointer, differing from it in having its verniers movable and its arcs fixed, instead of the opposite. It consists of a graduated circular arc fixed to the middle one of three long flat arms which turn about its centre, from which diverge their straight fiducial edges.

Fixed to each of the side arms is an index and vernier, by means of which those arms can be set so as to make any required angles with the middle arm. This instrument furnishes the readiest and most accurate graphic solution of the three-point problem on which hydrogra-phers so universally depend for determining positions of the sounding boat. The reflecting protractor, invented in January, 1874, by T. J. Lowry of the United States coast survey, enables one observer to measure at the same instant two adjacent angles, and plot them with the same instrument. It is obtained by placing between the fixed and each of the movable arms of the three-arm protractor an index arm; and each of these is so connected with those by means of jointed parallelograms that it always bisects the angle contained by the fixed arm and its corresponding movable protractor arm. Each of these index arms carries a mirror mounted perpendicular to its plane (and over its centre) of motion; these mirrors may be mounted to move either in the same or in parallel planes. (See Sextant.) Slightly forward of these mirrors on the line of sight is fixed a horizon glass, half silvered to admit of direct and reflected vision.

As the angular distance moved over by a mirror while measuring an angle is only half of the actual angle measured, and as each of these movable protractor arms is driven along its arc simultaneously with and twice as fast as its corresponding index arm, the angles contained by the fixed and movable protractor arms are the actual angles measured. When using the reflecting protractor the observer brings its face into the plane passing through his eye and three objects, and then sets his index arm so that the reflected and direct images of the objects (say left-hand and middle) of one of the desired angles are not coincident, yet approaching on account of the progress of the boat, and with the second index glass he makes the images of the right-hand and middle objects coincident, and keeps them so with the tangent screw till the first two objects become coincident, then clamps, and the angles are measured and also ready set off on the instrument. He now places the instrument on the map and shifts it until the fiducial edges of its protractor arms traverse the three points observed on, and dots the centre of the position.